# Differential Equations

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Paper 2, Section I, A

commentLet $y_{1}$ and $y_{2}$ be two linearly independent solutions to the differential equation

$\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+p(x) \frac{\mathrm{d} y}{\mathrm{~d} x}+q(x) y=0 .$

Show that the Wronskian $W=y_{1} y_{2}^{\prime}-y_{2} y_{1}^{\prime}$ satisfies

$\frac{\mathrm{d} W}{\mathrm{~d} x}+p W=0 .$

Deduce that if $y_{2}\left(x_{0}\right)=0$ then

$y_{2}(x)=y_{1}(x) \int_{x_{0}}^{x} \frac{W(t)}{y_{1}(t)^{2}} \mathrm{~d} t .$

Given that $y_{1}(x)=x^{3}$ satisfies the equation

$x^{2} \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}-x \frac{\mathrm{d} y}{\mathrm{~d} x}-3 y=0$

find the solution which satisfies $y(1)=0$ and $y^{\prime}(1)=1$.

Paper 2, Section I, A

commentSolve the difference equation

$y_{n+2}-4 y_{n+1}+4 y_{n}=n$

subject to the initial conditions $y_{0}=1$ and $y_{1}=0$.

Paper 2, Section II, A

commentBy means of the change of variables $\eta=x-t$ and $\xi=x+t$, show that the wave equation for $u=u(x, t)$

$\frac{\partial^{2} u}{\partial x^{2}}-\frac{\partial^{2} u}{\partial t^{2}}=0$

is equivalent to the equation

$\frac{\partial^{2} U}{\partial \eta \partial \xi}=0$

where $U(\eta, \xi)=u(x, t)$. Hence show that the solution to $(*)$ on $x \in \mathbf{R}$ and $t>0$, subject to the initial conditions

$u(x, 0)=f(x), \quad \frac{\partial u}{\partial t}(x, 0)=g(x)$

$u(x, t)=\frac{1}{2}[f(x-t)+f(x+t)]+\frac{1}{2} \int_{x-t}^{x+t} g(y) \mathrm{d} y$

Deduce that if $f(x)=0$ and $g(x)=0$ on the interval $\left|x-x_{0}\right|>r$ then $u(x, t)=0$ on $\left|x-x_{0}\right|>r+t$.

Suppose now that $y=y(x, t)$ is a solution to the wave equation $(*)$ on the finite interval $0<x<L$ and obeys the boundary conditions

$y(0, t)=y(L, t)=0$

for all $t$. The energy is defined by

$E(t)=\frac{1}{2} \int_{0}^{L}\left[\left(\frac{\partial y}{\partial x}\right)^{2}+\left(\frac{\partial y}{\partial t}\right)^{2}\right] \mathrm{d} x$

By considering $\mathrm{d} E / \mathrm{d} t$, or otherwise, show that the energy remains constant in time.

Paper 2, Section II, A

commentFor a linear, second order differential equation define the terms ordinary point, singular point and regular singular point.

For $a, b \in \mathbb{R}$ and $b \notin \mathbb{Z}$ consider the following differential equation

$x \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+(b-x) \frac{\mathrm{d} y}{\mathrm{~d} x}-a y=0 .$

Find coefficients $c_{m}(a, b)$ such that the function $y_{1}=M(x, a, b)$, where

$M(x, a, b)=\sum_{m=0}^{\infty} c_{m}(a, b) x^{m}$

satisfies $(*)$. By making the substitution $y=x^{1-b} u(x)$, or otherwise, find a second linearly independent solution of the form $y_{2}=x^{1-b} M(x, \alpha, \beta)$ for suitable $\alpha, \beta$.

Suppose now that $b=1$. By considering a limit of the form

$\lim _{b \rightarrow 1} \frac{y_{2}-y_{1}}{b-1}$

or otherwise, obtain two linearly independent solutions to $(*)$ in terms of $M$ and derivatives thereof.

Paper 2, Section II, A

commentFor an $n \times n$ matrix $A$, define the matrix exponential by

$\exp (A)=\sum_{m=0}^{\infty} \frac{A^{m}}{m !}$

where $A^{0} \equiv I$, with $I$ being the $n \times n$ identity matrix. [You may assume that $\exp ((s+t) A)=\exp (s A) \exp (t A)$ for real numbers $s, t$ and you do not need to consider issues of convergence.] Show that

$\frac{\mathrm{d}}{\mathrm{d} t} \exp (t A)=A \exp (t A)$

Deduce that the unique solution to the initial value problem

$\frac{\mathrm{d} \mathbf{y}}{\mathrm{d} t}=A \mathbf{y}, \quad \mathbf{y}(0)=\mathbf{y}_{0}, \quad \text { where } \mathbf{y}(t)=\left(\begin{array}{c} y_{1}(t) \\ \vdots \\ y_{n}(t) \end{array}\right)$

is $\mathbf{y}(t)=\exp (t A) \mathbf{y}_{0}$.

Let $\mathbf{x}=\mathbf{x}(t)$ and $\mathbf{f}=\mathbf{f}(t)$ be vectors of length $n$ and $A$ a real $n \times n$ matrix. By considering a suitable integrating factor, show that the unique solution to

$\frac{\mathrm{d} \mathbf{x}}{\mathrm{d} t}-A \mathbf{x}=\mathbf{f}, \quad \mathbf{x}(0)=\mathbf{x}_{0}$

is given by

$\mathbf{x}(t)=\exp (t A) \mathbf{x}_{0}+\int_{0}^{t} \exp [(t-s) A] \mathbf{f}(s) \mathrm{d} s$

Hence, or otherwise, solve the system of differential equations $(*)$ when

$A=\left(\begin{array}{ccc} 2 & 2 & -2 \\ 5 & 1 & -3 \\ 1 & 5 & -3 \end{array}\right), \quad \mathbf{f}(t)=\left(\begin{array}{c} \sin t \\ 3 \sin t \\ 0 \end{array}\right), \quad \mathbf{x}_{0}=\left(\begin{array}{l} 1 \\ 1 \\ 2 \end{array}\right)$

[Hint: Compute $A^{2}$ and show that $\left.A^{3}=0 .\right]$

Paper 2, Section II, A

commentThe function $\theta=\theta(t)$ takes values in the interval $(-\pi, \pi]$ and satisfies the differential equation

$\frac{\mathrm{d}^{2} \theta}{\mathrm{d} t^{2}}+(\lambda-2 \mu) \sin \theta+\frac{2 \mu \sin \theta}{\sqrt{5+4 \cos \theta}}=0$

where $\lambda$ and $\mu$ are positive constants.

Let $\omega=\dot{\theta}$. Express $(*)$ in terms of a pair of first order differential equations in $(\theta, \omega)$. Show that if $3 \lambda<4 \mu$ then there are three fixed points in the region $0 \leqslant \theta \leqslant \pi .$

Classify all the fixed points of the system in the case $3 \lambda<4 \mu$. Sketch the phase portrait in the case $\lambda=1$ and $\mu=3 / 2$.

Comment briefly on the case when $3 \lambda>4 \mu$.

Paper 1, Section I, A

commentSolve the differential equation

$\frac{d y}{d x}=\frac{1}{x+e^{2 y}}$

subject to the initial condition $y(1)=0$.

Paper 1, Section II, A

commentSolve the system of differential equations for $x(t), y(t), z(t)$,

$\begin{aligned} &\dot{x}=3 z-x \\ &\dot{y}=3 x+2 y-3 z+\cos t-2 \sin t \\ &\dot{z}=3 x-z \end{aligned}$

subject to the initial conditions $x(0)=y(0)=0, z(0)=1$.

Paper 1, Section II, A

commentShow that for each $t>0$ and $x \in \mathbb{R}$ the function

$K(x, t)=\frac{1}{\sqrt{4 \pi t}} \exp \left(-\frac{x^{2}}{4 t}\right)$

satisfies the heat equation

$\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}$

For $t>0$ and $x \in \mathbb{R}$ define the function $u=u(x, t)$ by the integral

$u(x, t)=\int_{-\infty}^{\infty} K(x-y, t) f(y) d y$

Show that $u$ satisfies the heat equation and $\lim _{t \rightarrow 0^{+}} u(x, t)=f(x)$. [Hint: You may find it helpful to consider the substitution $Y=(x-y) / \sqrt{4 t}$.]

Burgers' equation is

$\frac{\partial w}{\partial t}+w \frac{\partial w}{\partial x}=\frac{\partial^{2} w}{\partial x^{2}}$

By considering the transformation

$w(x, t)=-2 \frac{1}{u} \frac{\partial u}{\partial x}$

solve Burgers' equation with the initial condition $\lim _{t \rightarrow 0^{+}} w(x, t)=g(x)$.

Paper 2, Section I, $2 \mathrm{C}$

commentConsider the first order system

$\frac{d \boldsymbol{v}}{d t}-B \boldsymbol{v}=e^{\lambda t} \boldsymbol{x}$

to be solved for $\boldsymbol{v}(t)=\left(v_{1}(t), v_{2}(t), \ldots, v_{n}(t)\right) \in \mathbb{R}^{n}$, where the $n \times n$ matrix $B, \lambda \in \mathbb{R}$ and $\boldsymbol{x} \in \mathbb{R}^{n}$ are all independent of time. Show that if $\lambda$ is not an eigenvalue of $B$ then there is a solution of the form $\boldsymbol{v}(t)=e^{\lambda t} \boldsymbol{u}$, with $\boldsymbol{u}$ constant.

For $n=2$, given

$B=\left(\begin{array}{ll} 0 & 3 \\ 1 & 0 \end{array}\right) \quad \lambda=2 \quad \text { and } \boldsymbol{x}=\left(\begin{array}{l} 0 \\ 1 \end{array}\right)$

find the general solution to (1).

Paper 2, Section I, C

commentThe function $y(x)$ satisfies the inhomogeneous second-order linear differential equation

$y^{\prime \prime}-2 y^{\prime}-3 y=-16 x e^{-x}$

Find the solution that satisfies the conditions that $y(0)=1$ and $y(x)$ is bounded as $x \rightarrow \infty$.

Paper 2, Section II, $5 \mathrm{C}$

commentConsider the problem of solving

$\frac{d^{2} y}{d t^{2}}=t$

subject to the initial conditions $y(0)=\frac{d y}{d t}(0)=0$ using a discrete approach where $y$ is computed at discrete times, $y_{n}=y\left(t_{n}\right)$ where $t_{n}=n h(n=-1,0,1, \ldots, N)$ and $0<h=1 / N \ll 1 .$

(a) By using Taylor expansions around $t_{n}$, derive the centred-difference formula

$\frac{y_{n+1}-2 y_{n}+y_{n-1}}{h^{2}}=\left.\frac{d^{2} y}{d t^{2}}\right|_{t=t_{n}}+O\left(h^{\alpha}\right)$

where the value of $\alpha$ should be found.

(b) Find the general solution of $y_{n+1}-2 y_{n}+y_{n-1}=0$ and show that this is the discrete version of the corresponding general solution to $\frac{d^{2} y}{d t^{2}}=0$.

(c) The fully discretized version of the differential equation (1) is

$\frac{y_{n+1}-2 y_{n}+y_{n-1}}{h^{2}}=n h \quad \text { for } \quad n=0, \ldots, N-1$

By finding a particular solution first, write down the general solution to the difference equation (2). For the solution which satisfies the discretized initial conditions $y_{0}=0$ and $y_{-1}=y_{1}$, find the error in $y_{N}$ in terms of $h$ only.

Paper 2, Section II, $6 \mathrm{C}$

commentFind all power series solutions of the form $y=\sum_{n=0}^{\infty} a_{n} x^{n}$ to the equation

$\left(1-x^{2}\right) y^{\prime \prime}-x y^{\prime}+\lambda^{2} y=0$

for $\lambda$ a real constant. [It is sufficient to give a recurrence relationship between coefficients.]

Impose the condition $y^{\prime}(0)=0$ and determine those values of $\lambda$ for which your power series gives polynomial solutions (i.e., $a_{n}=0$ for $n$ sufficiently large). Give the values of $\lambda$ for which the corresponding polynomials have degree less than 6 , and compute these polynomials. Hence, or otherwise, find a polynomial solution of

$\left(1-x^{2}\right) y^{\prime \prime}-x y^{\prime}+y=8 x^{4}-3$

satisfying $y^{\prime}(0)=0$.

Paper 2, Section II, C

commentConsider the nonlinear system

$\begin{aligned} &\dot{x}=y-2 y^{3} \\ &\dot{y}=-x \end{aligned}$

(a) Show that $H=H(x, y)=x^{2}+y^{2}-y^{4}$ is a constant of the motion.

(b) Find all the critical points of the system and analyse their stability. Sketch the phase portrait including the special contours with value $H(x, y)=\frac{1}{4}$.

(c) Find an explicit expression for $y=y(t)$ in the solution which satisfies $(x, y)=\left(\frac{1}{2}, 0\right)$ at $t=0$. At what time does it reach the point $(x, y)=\left(\frac{1}{4},-\frac{1}{2}\right) ?$

Paper 2, Section II, C

commentTwo cups of tea at temperatures $T_{1}(t)$ and $T_{2}(t)$ cool in a room at ambient constant temperature $T_{\infty}$. Initially $T_{1}(0)=T_{2}(0)=T_{0}>T_{\infty}$.

Cup 1 has cool milk added instantaneously at $t=1$ and then hot water added at a constant rate after $t=2$ which is modelled as follows

$\frac{d T_{1}}{d t}=-a\left(T_{1}-T_{\infty}\right)-\delta(t-1)+H(t-2)$

whereas cup 2 is left undisturbed and evolves as follows

$\frac{d T_{2}}{d t}=-a\left(T_{2}-T_{\infty}\right)$

where $\delta(t)$ and $H(t)$ are the Dirac delta and Heaviside functions respectively, and $a$ is a positive constant.

(a) Derive expressions for $T_{1}(t)$ when $0<t \leqslant 1$ and for $T_{2}(t)$ when $t>0$.

(b) Show for $1<t<2$ that

$T_{1}(t)=T_{\infty}+\left(T_{0}-T_{\infty}-e^{a}\right) e^{-a t}$

(c) Derive an expression for $T_{1}(t)$ for $t>2$.

(d) At what time $t^{*}$ is $T_{1}=T_{2}$ ?

(e) Find how $t^{*}$ behaves for $a \rightarrow 0$ and explain your result.

Paper 2, Section I, B

commentShow that for given $P(x, y), Q(x, y)$ there is a function $F(x, y)$ such that, for any function $y(x)$,

$P(x, y)+Q(x, y) \frac{d y}{d x}=\frac{d}{d x} F(x, y)$

if and only if

$\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}$

Now solve the equation

$(2 y+3 x) \frac{d y}{d x}+4 x^{3}+3 y=0$

Paper 2, Section I, B

commentConsider the following difference equation for real $u_{n}$ :

$u_{n+1}=a u_{n}\left(1-u_{n}^{2}\right)$

where $a$ is a real constant.

For $-\infty<a<\infty$ find the steady-state solutions, i.e. those with $u_{n+1}=u_{n}$ for all $n$, and determine their stability, making it clear how the number of solutions and the stability properties vary with $a$. [You need not consider in detail particular values of $a$ which separate intervals with different stability properties.]

Paper 2, Section II, B

commentThe function $u(x, y)$ satisfies the partial differential equation

$a \frac{\partial^{2} u}{\partial x^{2}}+b \frac{\partial^{2} u}{\partial x \partial y}+c \frac{\partial^{2} u}{\partial y^{2}}=0$

where $a, b$ and $c$ are non-zero constants.

Defining the variables $\xi=\alpha x+y$ and $\eta=\beta x+y$, where $\alpha$ and $\beta$ are constants, and writing $v(\xi, \eta)=u(x, y)$ show that

$a \frac{\partial^{2} u}{\partial x^{2}}+b \frac{\partial^{2} u}{\partial x \partial y}+c \frac{\partial^{2} u}{\partial y^{2}}=A(\alpha, \beta) \frac{\partial^{2} v}{\partial \xi^{2}}+B(\alpha, \beta) \frac{\partial^{2} v}{\partial \xi \partial \eta}+C(\alpha, \beta) \frac{\partial^{2} v}{\partial \eta^{2}},$

where you should determine the functions $A(\alpha, \beta), B(\alpha, \beta)$ and $C(\alpha, \beta)$.

If the quadratic $a s^{2}+b s+c=0$ has distinct real roots then show that $\alpha$ and $\beta$ can be chosen such that $A(\alpha, \beta)=C(\alpha, \beta)=0$ and $B(\alpha, \beta) \neq 0$.

If the quadratic $a s^{2}+b s+c=0$ has a repeated root then show that $\alpha$ and $\beta$ can be chosen such that $A(\alpha, \beta)=B(\alpha, \beta)=0$ and $C(\alpha, \beta) \neq 0$.

Hence find the general solutions of the equations

$\frac{\partial^{2} u}{\partial x^{2}}+3 \frac{\partial^{2} u}{\partial x \partial y}+2 \frac{\partial^{2} u}{\partial y^{2}}=0$

and

$\frac{\partial^{2} u}{\partial x^{2}}+2 \frac{\partial^{2} u}{\partial x \partial y}+\frac{\partial^{2} u}{\partial y^{2}}=0$

Paper 2, Section II, B

commentBy choosing a suitable basis, solve the equation

$\left(\begin{array}{ll} 1 & 2 \\ 1 & 0 \end{array}\right)\left(\begin{array}{l} \dot{x} \\ \dot{y} \end{array}\right)+\left(\begin{array}{cc} -2 & 5 \\ 2 & -1 \end{array}\right)\left(\begin{array}{l} x \\ y \end{array}\right)=e^{-4 t}\left(\begin{array}{c} 3 b \\ 2 \end{array}\right)+e^{-t}\left(\begin{array}{c} -3 \\ c-1 \end{array}\right)$

subject to the initial conditions $x(0)=0, y(0)=0$.

Explain briefly what happens in the cases $b=2$ or $c=2$.

Paper 2, Section II, B

commentThe temperature $T$ in an oven is controlled by a heater which provides heat at rate $Q(t)$. The temperature of a pizza in the oven is $U$. Room temperature is the constant value $T_{r}$.

$T$ and $U$ satisfy the coupled differential equations

$\begin{aligned} \frac{d T}{d t} &=-a\left(T-T_{r}\right)+Q(t) \\ \frac{d U}{d t} &=-b(U-T) \end{aligned}$

where $a$ and $b$ are positive constants. Briefly explain the various terms appearing in the above equations.

Heating may be provided by a short-lived pulse at $t=0$, with $Q(t)=Q_{1}(t)=\delta(t)$ or by constant heating over a finite period $0<t<\tau$, with $Q(t)=Q_{2}(t)=\tau^{-1}(H(t)-H(t-$ $\tau)$, where $\delta(t)$ and $H(t)$ are respectively the Dirac delta function and the Heaviside step function. Again briefly, explain how the given formulae for $Q_{1}(t)$ and $Q_{2}(t)$ are consistent with their description and why the total heat supplied by the two heating protocols is the same.

For $t<0, T=U=T_{r}$. Find the solutions for $T(t)$ and $U(t)$ for $t>0$, for each of $Q(t)=Q_{1}(t)$ and $Q(t)=Q_{2}(t)$, denoted respectively by $T_{1}(t)$ and $U_{1}(t)$, and $T_{2}(t)$ and $U_{2}(t)$. Explain clearly any assumptions that you make about continuity of the solutions in time.

Show that the solutions $T_{2}(t)$ and $U_{2}(t)$ tend respectively to $T_{1}(t)$ and $U_{1}(t)$ in the limit as $\tau \rightarrow 0$ and explain why.

Paper 2, Section II, B

commentConsider the differential equation

$x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}-\left(x^{2}+\alpha^{2}\right) y=0$

What values of $x$ are ordinary points of the differential equation? What values of $x$ are singular points of the differential equation, and are they regular singular points or irregular singular points? Give clear definitions of these terms to support your answers.

For $\alpha$ not equal to an integer there are two linearly independent power series solutions about $x=0$. Give the forms of the two power series and the recurrence relations that specify the relation between successive coefficients. Give explicitly the first three terms in each power series.

For $\alpha$ equal to an integer explain carefully why the forms you have specified do not give two linearly independent power series solutions. Show that for such values of $\alpha$ there is (up to multiplication by a constant) one power series solution, and give the recurrence relation between coefficients. Give explicitly the first three terms.

If $y_{1}(x)$ is a solution of the above second-order differential equation then

$y_{2}(x)=y_{1}(x) \int_{c}^{x} \frac{1}{s\left[y_{1}(s)\right]^{2}} d s$

where $c$ is an arbitrarily chosen constant, is a second solution that is linearly independent of $y_{1}(x)$. For the case $\alpha=1$, taking $y_{1}(x)$ to be a power series, explain why the second solution $y_{2}(x)$ is not a power series.

[You may assume that any power series you use are convergent.]

Paper 2, Section I, $2 \mathrm{C}$

commentConsider the function

$f(x, y)=\frac{x}{y}+\frac{y}{x}-\frac{(x-y)^{2}}{a^{2}}$

defined for $x>0$ and $y>0$, where $a$ is a non-zero real constant. Show that $(\lambda, \lambda)$ is a stationary point of $f$ for each $\lambda>0$. Compute the Hessian and its eigenvalues at $(\lambda, \lambda)$.

Paper 2, Section I, C

comment(a) The numbers $z_{1}, z_{2}, \ldots$ satisfy

$z_{n+1}=z_{n}+c_{n} \quad(n \geqslant 1),$

where $c_{1}, c_{2}, \ldots$ are given constants. Find $z_{n+1}$ in terms of $c_{1}, c_{2}, \ldots, c_{n}$ and $z_{1}$.

(b) The numbers $x_{1}, x_{2}, \ldots$ satisfy

$x_{n+1}=a_{n} x_{n}+b_{n} \quad(n \geqslant 1),$

where $a_{1}, a_{2}, \ldots$ are given non-zero constants and $b_{1}, b_{2}, \ldots$ are given constants. Let $z_{1}=x_{1}$ and $z_{n+1}=x_{n+1} / U_{n}$, where $U_{n}=a_{1} a_{2} \cdots a_{n}$. Calculate $z_{n+1}-z_{n}$, and hence find $x_{n+1}$ in terms of $x_{1}, b_{1}, \ldots, b_{n}$ and $U_{1}, \ldots, U_{n}$.

Paper 2, Section II, $7 \mathrm{C}$

commentLet $y_{1}$ and $y_{2}$ be two solutions of the differential equation

$y^{\prime \prime}(x)+p(x) y^{\prime}(x)+q(x) y(x)=0, \quad-\infty<x<\infty$

where $p$ and $q$ are given. Show, using the Wronskian, that

either there exist $\alpha$ and $\beta$, not both zero, such that $\alpha y_{1}(x)+\beta y_{2}(x)$ vanishes for all $x$,

or given $x_{0}, A$ and $B$, there exist $a$ and $b$ such that $y(x)=a y_{1}(x)+b y_{2}(x)$ satisfies the conditions $y\left(x_{0}\right)=A$ and $y^{\prime}\left(x_{0}\right)=B$.

Find power series $y_{1}$ and $y_{2}$ such that an arbitrary solution of the equation

$y^{\prime \prime}(x)=x y(x)$

can be written as a linear combination of $y_{1}$ and $y_{2}$.

Paper 2, Section II, C

comment(a) Consider the system

$\begin{aligned} &\frac{d x}{d t}=x(1-x)-x y \\ &\frac{d y}{d t}=\frac{1}{8} y(4 x-1) \end{aligned}$

for $x(t) \geqslant 0, y(t) \geqslant 0$. Find the critical points, determine their type and explain, with the help of a diagram, the behaviour of solutions for large positive times $t$.

(b) Consider the system

$\begin{aligned} &\frac{d x}{d t}=y+\left(1-x^{2}-y^{2}\right) x \\ &\frac{d y}{d t}=-x+\left(1-x^{2}-y^{2}\right) y \end{aligned}$

for $(x(t), y(t)) \in \mathbb{R}^{2}$. Rewrite the system in polar coordinates by setting $x(t)=$ $r(t) \cos \theta(t)$ and $y(t)=r(t) \sin \theta(t)$, and hence describe the behaviour of solutions for large positive and large negative times.

Paper 2, Section II, C

commentThe current $I(t)$ at time $t$ in an electrical circuit subject to an applied voltage $V(t)$ obeys the equation

$L \frac{d^{2} I}{d t^{2}}+R \frac{d I}{d t}+\frac{1}{C} I=\frac{d V}{d t}$

where $R, L$ and $C$ are the constant resistance, inductance and capacitance of the circuit with $R \geqslant 0, L>0$ and $C>0$.

(a) In the case $R=0$ and $V(t)=0$, show that there exist time-periodic solutions of frequency $\omega_{0}$, which you should find.

(b) In the case $V(t)=H(t)$, the Heaviside function, calculate, subject to the condition

$R^{2}>\frac{4 L}{C}$

the current for $t \geqslant 0$, assuming it is zero for $t<0$.

(c) If $R>0$ and $V(t)=\sin \omega_{0} t$, where $\omega_{0}$ is as in part (a), show that there is a timeperiodic solution $I_{0}(t)$ of period $T=2 \pi / \omega_{0}$ and calculate its maximum value $I_{M}$.

(i) Calculate the energy dissipated in each period, i.e., the quantity

$D=\int_{0}^{T} R I_{0}(t)^{2} d t$

Show that the quantity defined by

$Q=\frac{2 \pi}{D} \times \frac{L I_{M}^{2}}{2}$

satisfies $Q \omega_{0} R C=1$.

(ii) Write down explicitly the general solution $I(t)$ for all $R>0$, and discuss the relevance of $I_{0}(t)$ to the large time behaviour of $I(t)$.

Paper 2, Section II, C

comment(a) Solve $\frac{d z}{d t}=z^{2}$ subject to $z(0)=z_{0}$. For which $z_{0}$ is the solution finite for all $t \in \mathbb{R}$ ?

Let $a$ be a positive constant. By considering the lines $y=a\left(x-x_{0}\right)$ for constant $x_{0}$, or otherwise, show that any solution of the equation

$\frac{\partial f}{\partial x}+a \frac{\partial f}{\partial y}=0$

is of the form $f(x, y)=F(y-a x)$ for some function $F$.

Solve the equation

$\frac{\partial f}{\partial x}+a \frac{\partial f}{\partial y}=f^{2}$

subject to $f(0, y)=g(y)$ for a given function $g$. For which $g$ is the solution bounded on $\mathbb{R}^{2}$ ?

(b) By means of the change of variables $X=\alpha x+\beta y$ and $T=\gamma x+\delta y$ for appropriate real numbers $\alpha, \beta, \gamma, \delta$, show that the equation

$\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial x \partial y}=0$

can be transformed into the wave equation

$\frac{1}{c^{2}} \frac{\partial^{2} F}{\partial T^{2}}-\frac{\partial^{2} F}{\partial X^{2}}=0$

where $F$ is defined by $f(x, y)=F(\alpha x+\beta y, \gamma x+\delta y)$. Hence write down the general solution of $(*)$.

Paper 2, Section $I$, A

comment(a) For each non-negative integer $n$ and positive constant $\lambda$, let

$I_{n}(\lambda)=\int_{0}^{\infty} x^{n} e^{-\lambda x} d x$

By differentiating $I_{n}$ with respect to $\lambda$, find its value in terms of $n$ and $\lambda$.

(b) By making the change of variables $x=u+v, y=u-v$, transform the differential equation

$\frac{\partial^{2} f}{\partial x \partial y}=1$

into a differential equation for $g$, where $g(u, v)=f(x, y)$.

Paper 2, Section I, A

comment(a) Find the solution of the differential equation

$y^{\prime \prime}-y^{\prime}-6 y=0$

that is bounded as $x \rightarrow \infty$ and satisfies $y=1$ when $x=0$.

(b) Solve the difference equation

$\left(y_{n+1}-2 y_{n}+y_{n-1}\right)-\frac{h}{2}\left(y_{n+1}-y_{n-1}\right)-6 h^{2} y_{n}=0 .$

Show that if $0<h \ll 1$, the solution that is bounded as $n \rightarrow \infty$ and satisfies $y_{0}=1$ is approximately $(1-2 h)^{n}$.

(c) By setting $x=n h$, explain the relation between parts (a) and (b).

Paper 2, Section II, $6 A$

comment(a) The function $y(x)$ satisfies

$y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0$

(i) Define the Wronskian $W(x)$ of two linearly independent solutions $y_{1}(x)$ and $y_{2}(x)$. Derive a linear first-order differential equation satisfied by $W(x)$.

(ii) Suppose that $y_{1}(x)$ is known. Use the Wronskian to write down a first-order differential equation for $y_{2}(x)$. Hence express $y_{2}(x)$ in terms of $y_{1}(x)$ and $W(x)$.

(b) Verify that $y_{1}(x)=\cos \left(x^{\gamma}\right)$ is a solution of

$a x^{\alpha} y^{\prime \prime}+b x^{\alpha-1} y^{\prime}+y=0,$

where $a, b, \alpha$ and $\gamma$ are constants, provided that these constants satisfy certain conditions which you should determine.

Use the method that you described in part (a) to find a solution which is linearly independent of $y_{1}(x)$.

Paper 2, Section II, A

comment(a) Find and sketch the solution of

$y^{\prime \prime}+y=\delta(x-\pi / 2)$

where $\delta$ is the Dirac delta function, subject to $y(0)=1$ and $y^{\prime}(0)=0$.

(b) A bowl of soup, which Sam has just warmed up, cools down at a rate equal to the product of a constant $k$ and the difference between its temperature $T(t)$ and the temperature $T_{0}$ of its surroundings. Initially the soup is at temperature $T(0)=\alpha T_{0}$, where $\alpha>2$.

(i) Write down and solve the differential equation satisfied by $T(t)$.

(ii) At time $t_{1}$, when the temperature reaches half of its initial value, Sam quickly adds some hot water to the soup, so the temperature increases instantaneously by $\beta$, where $\beta>\alpha T_{0} / 2$. Find $t_{1}$ and $T(t)$ for $t>t_{1}$.

(iii) Sketch $T(t)$ for $t>0$.

(iv) Sam wants the soup to be at temperature $\alpha T_{0}$ at time $t_{2}$, where $t_{2}>t_{1}$. What value of $\beta$ should Sam choose to achieve this? Give your answer in terms of $\alpha$, $k, t_{2}$ and $T_{0}$.

Paper 2, Section II, A

comment(a) By considering eigenvectors, find the general solution of the equations

$\tag{†} \begin{aligned} &\frac{d x}{d t}=2 x+5 y \\ &\frac{d y}{d t}=-x-2 y \end{aligned}$

and show that it can be written in the form

$\left(\begin{array}{l} x \\ y \end{array}\right)=\alpha\left(\begin{array}{c} 5 \cos t \\ -2 \cos t-\sin t \end{array}\right)+\beta\left(\begin{array}{c} 5 \sin t \\ \cos t-2 \sin t \end{array}\right)$

where $\alpha$ and $\beta$ are constants.

(b) For any square matrix $M$, $\exp (M)$ is defined by

$\exp (M)=\sum_{n=0}^{\infty} \frac{M^{n}}{n !}$

Show that if $M$ has constant elements, the vector equation $\frac{d \mathbf{x}}{d t}=M \mathbf{x}$ has a solution $\mathbf{x}=\exp (M t) \mathbf{x}_{0}$, where $\mathbf{x}_{0}$ is a constant vector. Hence solve $(†)$ and show that your solution is consistent with the result of part (a).

Paper 2, Section II, A

commentThe function $y(x)$ satisfies

$y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0$

What does it mean to say that the point $x=0$ is (i) an ordinary point and (ii) a regular singular point of this differential equation? Explain what is meant by the indicial equation at a regular singular point. What can be said about the nature of the solutions in the neighbourhood of a regular singular point in the different cases that arise according to the values of the roots of the indicial equation?

State the nature of the point $x=0$ of the equation

$x y^{\prime \prime}+(x-m+1) y^{\prime}-(m-1) y=0$

Set $y(x)=x^{\sigma} \sum_{n=0}^{\infty} a_{n} x^{n}$, where $a_{0} \neq 0$, and find the roots of the indicial equation.

(a) Show that one solution of $(*)$ with $m \neq 0,-1,-2, \cdots$ is

$y(x)=x^{m}\left(1+\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{n}}{(m+n)(m+n-1) \cdots(m+1)}\right)$

and find a linearly independent solution in the case when $m$ is not an integer.

(b) If $m$ is a positive integer, show that $(*)$ has a polynomial solution.

(c) What is the form of the general solution of $(*)$ in the case $m=0$ ? [You do not need to find the general solution explicitly.]

Paper 2, Section $I$, B

commentFind the general solution of the equation

$2 \frac{d y}{d t}=y-y^{3} .$

Compute all possible limiting values of $y$ as $t \rightarrow \infty$.

Find a non-zero value of $y(0)$ such that $y(t)=y(0)$ for all $t$.

Paper 2, Section I, B

commentFind the general solution of the equation

$\frac{d y}{d x}-2 y=e^{\lambda x}$

where $\lambda$ is a constant not equal to 2 .

By subtracting from the particular integral an appropriate multiple of the complementary function, obtain the limit as $\lambda \rightarrow 2$ of the general solution of $(*)$ and confirm that it yields the general solution for $\lambda=2$.

Solve equation $(*)$ with $\lambda=2$ and $y(1)=2$.

Paper 2, Section II, B

commentConsider the equation

$2 \frac{\partial^{2} u}{\partial x^{2}}+3 \frac{\partial^{2} u}{\partial y^{2}}-7 \frac{\partial^{2} u}{\partial x \partial y}=0$

for the function $u(x, y)$, where $x$ and $y$ are real variables. By using the change of variables

$\xi=x+\alpha y, \quad \eta=\beta x+y$

where $\alpha$ and $\beta$ are appropriately chosen integers, transform $(*)$ into the equation

$\frac{\partial^{2} u}{\partial \xi \partial \eta}=0$

Hence, solve equation $(*)$ supplemented with the boundary conditions

$u(0, y)=4 y^{2}, \quad u(-2 y, y)=0, \quad \text { for all } y$

Paper 2, Section II, B

commentWrite as a system of two first-order equations the second-order equation

$\frac{d^{2} \theta}{d t^{2}}+c \frac{d \theta}{d t}\left|\frac{d \theta}{d t}\right|+\sin \theta=0$

where $c$ is a small, positive constant, and find its equilibrium points. What is the nature of these points?

Draw the trajectories in the $(\theta, \omega)$ plane, where $\omega=d \theta / d t$, in the neighbourhood of two typical equilibrium points.

By considering the cases of $\omega>0$ and $\omega<0$ separately, find explicit expressions for $\omega^{2}$ as a function of $\theta$. Discuss how the second term in $(*)$ affects the nature of the equilibrium points.

Paper 2, Section II, B

commentSuppose that $\mathbf{x}(t) \in \mathbb{R}^{3}$ obeys the differential equation

$\frac{d \mathbf{x}}{d t}=M \mathbf{x}$

where $M$ is a constant $3 \times 3$ real matrix.

(i) Suppose that $M$ has distinct eigenvalues $\lambda_{1}, \lambda_{2}, \lambda_{3}$ with corresponding eigenvectors $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$. Explain why $\mathbf{x}$ may be expressed in the form $\sum_{i=1}^{3} a_{i}(t) \mathbf{e}_{i}$ and deduce by substitution that the general solution of $(*)$ is

$\mathbf{x}=\sum_{i=1}^{3} A_{i} e^{\lambda_{i} t} \mathbf{e}_{i}$

where $A_{1}, A_{2}, A_{3}$ are constants.

(ii) What is the general solution of $(*)$ if $\lambda_{2}=\lambda_{3} \neq \lambda_{1}$, but there are still three linearly independent eigenvectors?

(iii) Suppose again that $\lambda_{2}=\lambda_{3} \neq \lambda_{1}$, but now there are only two linearly independent eigenvectors: $\mathbf{e}_{1}$ corresponding to $\lambda_{1}$ and $\mathbf{e}_{2}$ corresponding to $\lambda_{2}$. Suppose that a vector $\mathbf{v}$ satisfying the equation $\left(M-\lambda_{2} I\right) \mathbf{v}=\mathbf{e}_{2}$ exists, where $I$ denotes the identity matrix. Show that $\mathbf{v}$ is linearly independent of $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$, and hence or otherwise find the general solution of $(*)$.

Paper 2, Section II, B

commentSuppose that $u(x)$ satisfies the equation

$\frac{d^{2} u}{d x^{2}}-f(x) u=0$

where $f(x)$ is a given non-zero function. Show that under the change of coordinates $x=x(t)$,

$\frac{d^{2} u}{d t^{2}}-\frac{\ddot{x}}{\dot{x}} \frac{d u}{d t}-\dot{x}^{2} f(x) u=0$

where a dot denotes differentiation with respect to $t$. Furthermore, show that the function

$U(t)=\dot{x}^{-\frac{1}{2}} u(x)$

satisfies

$\frac{d^{2} U}{d t^{2}}-\left[\dot{x}^{2} f(x)+\dot{x}^{-\frac{1}{2}}\left(\frac{\ddot{x}}{\dot{x}} \frac{d}{d t}\left(\dot{x}^{\frac{1}{2}}\right)-\frac{d^{2}}{d t^{2}}\left(\dot{x}^{\frac{1}{2}}\right)\right)\right] U=0$

Choosing $\dot{x}=(f(x))^{-\frac{1}{2}}$, deduce that

$\frac{d^{2} U}{d t^{2}}-(1+F(t)) U=0$

for some appropriate function $F(t)$. Assuming that $F$ may be neglected, deduce that $u(x)$ can be approximated by

$u(x) \approx A(x)\left(c_{+} e^{G(x)}+c_{-} e^{-G(x)}\right),$

where $c_{+}, c_{-}$are constants and $A, G$ are functions that you should determine in terms of $f(x)$.

Paper 2, Section I, B

commentConsider the ordinary differential equation

$P(x, y)+Q(x, y) \frac{d y}{d x}=0 .$

State an equation to be satisfied by $P$ and $Q$ that ensures that equation $(*)$ is exact. In this case, express the general solution of equation $(*)$ in terms of a function $F(x, y)$ which should be defined in terms of $P$ and $Q$.

Consider the equation

$\frac{d y}{d x}=-\frac{4 x+3 y}{3 x+3 y^{2}}$

satisfying the boundary condition $y(1)=2$. Find an explicit relation between $y$ and $x$.

Paper 2, Section I, B

commentThe following equation arises in the theory of elastic beams:

$t^{4} \frac{d^{2} u}{d t^{2}}+\lambda^{2} u=0, \quad \lambda>0, t>0$

where $u(t)$ is a real valued function.

By using the change of variables

$t=\frac{1}{\tau}, \quad u(t)=\frac{v(\tau)}{\tau},$

find the general solution of the above equation.

Paper 2, Section II, B

The so-called "shallow water theory" is characterised by the equations

$\begin{aligned} &\frac{\partial \zeta}{\partial t}+\frac{\partial}{\partial x}[(h+\zeta) u]=0 \\ &\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}+g \frac{\partial \zeta}{\partial x}=0 \end{aligned}$

where $g$ denotes the gravitational constant, the constant $h$ denotes the undisturbed depth of the water, $u(x, t)$ denotes the speed in the $x$